Weight of 2D shape

Question

I want to compute a weight of the yellow object on the picture.
The density of the object is a function $$p(x,y)=\frac{y}{x^2+y^2}$$

I want to compute it using volume of the yellow object and average value of a density function. (volume * average density) Is it possible to compute it that way?

Lot of thanks!

Answer 1

Hint: Sure. Let your yellow region be $Y$. We want $$\iint_Y \frac{y}{x^2+y^2}\,dy\,dx.$$ To evaluate, it is natural to switch to polar coordinates. We get $$\int_1^3\int_0^\pi \cos\theta\,d\theta\,dr.$$

Answer 2

As average density is mass/volume, then yes.

So you just have to integrate the density over the yellow region.

$\int_r^R \int_0^\pi sin(\theta) dr d\theta$